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Difference between revisions of "Rank of a Lie group"

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''(real or complex)''
  
The (real, respectively, complex) dimension of any [[Cartan subgroup|Cartan subgroup]] of it. The rank of a Lie group coincides with the rank of its Lie algebra (see [[Rank of a Lie algebra|Rank of a Lie algebra]]). If a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077460/r0774601.png" /> coincides with the set of real or complex points of a [[Linear algebraic group|linear algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077460/r0774602.png" />, then the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077460/r0774603.png" /> coincides with the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077460/r0774604.png" /> (cf. [[Rank of an algebraic group|Rank of an algebraic group]]).
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The (real, respectively, complex) dimension of any [[Cartan subgroup|Cartan subgroup]] of it. The rank of a Lie group coincides with the rank of its Lie algebra (see [[Rank of a Lie algebra|Rank of a Lie algebra]]). If a Lie group $G$ coincides with the set of real or complex points of a [[Linear algebraic group|linear algebraic group]] $\widehat G$, then the rank of $G$ coincides with the rank of $\widehat G$ (cf. [[Rank of an algebraic group|Rank of an algebraic group]]).
  
  

Revision as of 16:22, 19 April 2014

(real or complex)

The (real, respectively, complex) dimension of any Cartan subgroup of it. The rank of a Lie group coincides with the rank of its Lie algebra (see Rank of a Lie algebra). If a Lie group $G$ coincides with the set of real or complex points of a linear algebraic group $\widehat G$, then the rank of $G$ coincides with the rank of $\widehat G$ (cf. Rank of an algebraic group).


Comments

References

[a1] R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972) pp. Chapt. 1
[a2] V.S. Varadarajan, "Lie groups, Lie algebras, and their representations" , Prentice-Hall (1974) MR0376938 Zbl 0371.22001
[a3] A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Press (1986) MR0855239 Zbl 0604.22001
[a4] J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 MR0396773 Zbl 0325.20039
How to Cite This Entry:
Rank of a Lie group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rank_of_a_Lie_group&oldid=31877
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article